A counting example inspired by a Cristina Milos tweet

Saw this nice tweet from Cristina Milos earlier today:

It gets to something that I’ve been seeing with my kids that I don’t really understand. I’ll say right from the start that I may not have the situation exactly right, so take the next part with that in mind. But . . .

It seems to me that up until this year the boys would approach similar problems in a similar way. I was always struck that even with a 2 1/2 year age difference, their approaches were so much alike.

This year, though, my older son has started incorporating more sophisticated methods in his solutions. In the language in Milos’s tweet – he seems to have acquired (some of, of course) the schematic knowledge needed for efficient problem solving.

As I said above, I don’t know what caused him to start using these new approaches, but right now their two approaches to problem solving are noticeably different. I picked a problem that gave my older son a little trouble today to illustrate the effects I’m seeing.

The problem is from the 2005 AMC 8 and asks you to count the number of triangles that you can form from 6 dots:

Problem 21 from the 2005 AMC 8

My older son uses counting techniques and choosing numbers:


By comparison my younger son draws all 18 triangles:


There’s obviously nothing wrong with my younger son’s approach – and he even sorts the triangles into groups of congruent triangles, which is neat. His solution is less sophisticated, though. Since until very recently it seemed like the two kids had similar approaches to most problems, I’m interested to try to understand where the difference in approach suddenly came from.